To be more precise, suppose that \(p\)  is an irregular vertex of our initial network \(\Gamma _0\)  which is
\(k\) -valent for some \(k > 3\)  (the construction near vertices of valence 3 is slightly simpler). Assume
for simplicity \(p\)  is placed at the origin. Enumerate the curves meeting at \(p\)  by \(\gamma ^{(1)}, \ldots , \gamma ^{(k)}\) ,
sequentially going counterclockwise. Denote by \(\tau _j\)  the unit tangent vector to \(\gamma _j\)  at \(p\) , and let
\(\ell _j = \mathbb {R}^+ \tau _j\)  be the ray in \(\mathbb {R}^2\)  in the direction \(\tau _j\) . The incoming edges at \(p\) 
thus determine a fan of rays in \(\mathbb {R}^2\)  emanating from the origin. As proved in [1]}, there exists at
least one (and in fact, multiple when \(k > 3\) ) expanding soliton solutions for the curve-shortening flow
which have this fan as initial condition. (At least for generic fans, but conjecturally for all fans, there are only finitely many
such solitons.) Choose one of these solutions, and denote it by \(S_p\) . Having made this choice for every irregular
interior vertex, we prove that there exists a unique solution \(\Gamma (t)\)  of the network flow, defined on some interval
\(0 < t < T\) , whose combinatorics are the same as if we were to replace a small ball around each irregular vertex \(p\) 
with the corresponding choice of soliton \(S_p\) . This solution converges as \(t \rightarrow 0\)  to \(\Gamma _0\) , but
this convergence happens in a slightly odd fashion since \(\Gamma (t)\)  may have more curves than \(\Gamma _0\) ,
so certain curves in \(\Gamma (t)\)  converge to curves in \(\Gamma _0\)  (as smoothly as the regularity of those curves
in \(\Gamma _0\)  allow), while the remaining curves in \(\Gamma (t)\)  shrink in size and `disappear into' these
irregular vertices as \(t \searrow 0\) . There is a sense in which the entire convergence can be regarded
as \({\mathcal {C}}^\infty \)  if the initial network \(\Gamma _0\)  is smooth.
At each irregular vertex \(p\) , the soliton \(S_p\)  is the tangent flow at \((p,0)\) . This replacement of
a neighborhood of an irregular vertex \(p\)  with a soliton \(S_p\)  appears also in [2]}, but their
analysis only describes the convergence as \(t \searrow 0\)  in a much weaker sense.
